Two goals from Merouane Zemmama and one each for Andy Halliday and Ishmael Miller saw Boro into round four, but Hastings teenager Bradley Goldberg struck a superb consolation while midfielder Lee Carey hit a post.The Ryman Premier Division strugglers, ranked 131 places below their opponents in the league structure, trailed only to Zemmama's opener at the interval after keeper Liam O'Brien saved an early Miller penalty.
Mowbray said: "It was a credit to them that they tried to play football right from the off. I was pleased for them that they scored a great goal in front of their supporters. It was nice to be able to congratulate them afterwards.
"It was a bit patchy from us but FA Cups are about going through to the next round. It's important to keep the confidence and belief going no matter the name of the team you're playing. We got the job done, but we could have made it easier."
Sean Ray's underdogs weathered an early storm before keeper O'Brien, brought in from Barnet on Friday after an injury to regular number one and part-time naked butler Matt Armstrong-Ford, guessed right to save a 17th-minute penalty from Miller.
The visitors' day was capped by Goldberg's 25-yard strike which brought applause from all four sides of the ground and Hastings player-boss Ray said: "Seeing how happy our supporters were at the end is a memory I will savour for the rest of my life.
"I'm so pleased we were able to get a goal. Bradley has got that in his locker. He does it week in, week out and he is a threat whoever he plays against. He has got pace and he knows where the back of the net is.
"I thought were a bit overawed in the first half but we started to relax more and caused them one or two problems."
It was recalled defender Zemmama who finally grabbed the lead for Boro after 22 minutes when he curled the ball into the top corner from the edge of the box.
But Hastings hit back with Jamie Crelin having a shot blocked and then Goldberg hared through, rounded Boro keeper Jason Steele and put the ball in the net before being pulled back for a marginal offside.
And on the stroke of half-time Hastings had their best chance yet when Bailo Camara sent in a dangerous cross from the right and Halliday almost deflected the ball past his own keeper Steele, who parried it out for a corner.
If the visitors went in at the break still with hope, it looked to have been shattered just two minutes after the interval when Halliday prodded in Boro's second from a Luke Williams corner.
After Carey forced Steele into a superb one-handed save, Zemmama made it 3-0 to Boro in the 68th minute with a heavily deflected shot from the edge of the box.
But almost immediately after the restart, Hastings pushed forward through Goldberg, who bore down on goal before looking up and letting fly with a long-range effort which beat Steele and nestled in the top corner of the net.
Miller completed Boro's win five minutes from time, but Hastings took the plaudits and Ray added: "The boys have achieved so much this year and I couldn't have asked any more of them. They have done their town, their support and themselves proud."
Ray's men get back to reality on Tuesday with a Ryman League clash with Wealdstone, but they will put preparations on hold as they toast their fairytale FA Cup run with a party in Newcastle tonight.
"I'll start worrying about that one tomorrow," added Ray. "We're out in Newcastle tonight and we've already booked the nightclub. The boys deserve a night out to let their hair down. I might have one or two myself."
Full coverage of all the FA Cup fixtures at ITV's sports website here.